A sequence of Picard/Galois orbits of special points in a product of arbitrary many modular curves is conjectured to equidistribute in the product space as long as it escapes any closed orbit of an intermediate subgroup. This conjecture encompasses several well-known results and conjectures, including Duke's Theorem, the Michel-Venkatesh mixing conjecture and the equidistribution strengthening of André-Oort in this setting.
I will present an argument to prove that limit measure of orbits of special points are singular with respect to intermediate homogeneous measures. The method utilizes the geometric expansion of a relative trace and a sieve argument. We need also to assume that Dirichlet L-functions have no exceptional Landau-Siegel zero.
This lecture is a conceptual sequel to the talk on Tuesday but is formally independent of it.